I will introduce the idea of residence time distributions here. This is an important topic
in chemical reactors and it's important because not all molecules spend the same time in the
reactor. This is the thing to keep in mind and the question is why is that important?
It's important because different residence times mean different reaction times and this
can significantly affect the product distribution when we have more than one reaction taking
place and almost every system has more than one reaction so product selectivity, for example,
in polymerization reactors, polymer molecular weight distribution can be significantly affected
by the residence time distribution. So we'd like to be able to determine the residence
time distribution for a real reactor and we can do this experimentally using a tracer
technique. So if this represents our reactor, what we do is inject a tracer into the feed
side of the reactor. And then we have a detector here, this is the detector. What it's doing
is measuring the concentration of the tracer leaving the reactor. So we inject a tracer
at the inlet, what we call time zero. It takes some time for the tracer to move through the
reactor and because different molecules spend different amounts of time in the reactor in
general, then some of the tracer is going to show up sooner than other parts of the
tracer that was injected because of this residence time distribution. And the tracer can be a
dye, it can be a radioactive tracer material, it can be an electrolyte, something that is
going to have similar properties as the feed to the reactor. And so if we do this, what
do we expect the behavior to look like at the outlet? So here is the idea, that we inject
a tracer, so this injected tracer looks like a very narrow pulse where I'm plotting concentration
of the tracer as a function of time. We inject in a very short time. And then the outlet,
we're going to see something that might look like this. So the outlet detector, we're going
to get some tracer that's spread out in time. This is the outlet. And it's spread out in
time because different molecules spend different times in the reactor. Area under these curves,
so it's proportional to the amount of tracer, which means the area under these two curves
should be identical if our tracer did not adsorb on a catalyst for example, somehow
get held up in the system for a much longer time. So I'm going to take this part and blow
it up to create the view here. Then if we look at some small time difference, this area,
that's delta N where N is the number of moles of the tracer, concentration, this is the
total volumetric flow rate through the reactor, and this is the time interval that we picked.
The amount that we injected, we can call that N zero. It's going to be the integral of the concentration
over time from time zero to a very long time, times the volumetric flow rate. This then
is equal to the amount initially injected and we expect of course that all of it would
leave. So what's important here? This then corresponds to the amount of material that
is spent between time, t, and time t+dt in the reactor. Of course, we want to normalize
this because the properties of the reactor should not depend on how much tracer we use.
So deltaN divided by the amount that's injected then is going to give us the fraction of material
that has spent the time between t and t+dt in the reactor. So deltaN, the volumetric
flow rate, the concentration over that interval, delta t, divided by N zero. We're going to call
this p(t)delta t. Of course, what we're interested in is as delta t becomes small, it becomes
a differential, then we're going to have p(t)dt. This p(t) then is our residence time distribution,
also known as the exit age distribution. Namely, we're looking at the exit of the reactor and
asking how long the molecule has spent in the reactor, what's their age in the reactor.
And the p(t) is related to this concentration that we measure as a function of t. Volumetric
flow rate times that concentration over the total, 0 to infinity of the volumetric flow
rate*concentration. So this just becomes the concentration over this integral of concentration
with respect to time. So this p(t)dt is the fraction of material that has spent the residence
time between t and t+dt in the reactor. So p(t) is this probability the molecule will
reside in the reactor for time t which means that molecule will have that much time to
react. Of course, the longer molecules spend in the reactor, the more likely they are to
react. Now we can calculate these residence time distributions using probabilities for
ideal reactors, but we must measure them for real reactors using a tracer method like I
just described.